Applied Electronics I Bipolar Junction Transistors

CHAPTER 3
BIPOLAR JUNCTION TRANSISTORS

3.1. Introduction
Transistors are three-layered, three- terminal and two-junction electronic devices constructed of
doped semiconductor and whose voltage-current relationship is controlled by a third voltage or
current. We may regard a transistor as a controlled voltage or current source. The word transistor
is the combination of two words. TRANsfer and reSISTOR (transfer and resistor) which means
transfer of electrical power from a low resistive circuit to a high resistive circuit.

They were demonstrated by a team of scientists at Bell laboratories in 1947 and their
introduction brought an end to the age of vacuum tube devices due to some of their merits over
vacuum tubes such as:
• Smaller size, light weight
• No heating elements required
• Low power consumption
• Low operating voltages
Transistors are used in such applications as signal amplifiers, electronic switches, oscillators,
design of digital logics, memory circuits etc. Depending on their majority and minority charge
carriers, transistors can be classified as:

a. Bipolar Transistors b. Unipolar Transistors

Bipolar transistors are so named because their operation involves both electrons and holes.
Charge flow in such transistors is due to bidirectional diffusion of charge carriers across a
junction between two regions of different charge concentrations. Thus, in bipolar transistors the
charge carriers are electrons and holes (majority charge carriers and minority charge carriers)
they are principally called Bipolar Junction Transistors (BJTs).

In Unipolar transistors only one carrier type is involved in charge flow due to drift. This charge
carrier is either electrons or holes as majority charge carriers only. Since only one type of charge
is current carrier, such transistors are called Unipolar Junction Transistors (UJTs). The unipolar
junction transistors are mainly known as Field Effect Transistors (FETs). FETs are to be
discussed in the next chapter.

By design, most of the BJT collector current is due to the flow of charges injected from a high-
concentration emitter into the base where they are minority carriers that diffuse toward the
collector, and so BJTs are classified as minority-carrier devices or current controlled devices
while FETs are said to be voltage controlled devices.
3.2. Bipolar Junction Transistors (BJTs)

A BJT consists of three differently doped semiconductor regions namely; the emitter region, the

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base region and the collector region. These regions are, respectively, n type, p type and n type in
an NPN transistor , and p type, n type and p type in a PNP. Thus, there are two types of bipolar
junction transistor.

1. A thin layer of P-type material is sandwiched between two N-type materials which is then
known as an NPN transistor Fig.3.1 (a).
2. A thin layer of N-type material is sandwiched between two P-type materials to form a PNP
transistor Fig.3.1. (b).

(a) (b)

(c) (d)

Fig.3.1 BJT construction (a, b, c) and schematic symbols (d)

Each semiconductor region is connected to a terminal, appropriately labeled: emitter (E), base
(B) and collector (C). The arrow head on the emitter always indicates to the N-type region and to
the conventional current flow direction.

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In both NPN and PNP transistors constructions, the base region is physically located between the
emitter and the collector and is made from lightly doped, high resistive material. It allows most
of the charge carriers to pass through it from the emitter to the collector (current controlling).The
emitter region is usually of low resistive material, which is heavily doped and supplies majority
charge carriers. The collector region is doped slightly lower than the emitter region and it
collects the most majority charge carriers. Therefore, due to the above cases the depletion layers
penetrate into the base region (Fig.3.3) and a transistor is considered as a combination of two pn-
junction diodes (Fig.3.1c). In other words, we can see that there are two junctions shared
between the three terminals, the Emitter-base junction and Collector-base junction.

3.3. Principles of Operation

For their appropriate (correct) operation both NPN and PNP transistors must be properly biased.

Biasing can be defined as a dc voltage and current that is applied to an electronics device to set
up the desired dc operating points.

The operating point of a device, also known as bias point, quiescent point, or Q-point, is the
point on the output characteristics that shows the DC collector–emitter voltage (Vce) and the
collector current (Ic) with no ac input signal applied. The term is normally used in connection
with devices such as transistors under their dc conditions.

Generally, there are four different junction-biasing combinations to have four distinct regions of
operation.

Regions of Operation

The modes of operation can be described in terms of the junction biasing:

 Forward-Active (or simply, Active): The base–emitter junction is forward biased and
the base–collector junction is reverse biased. Most bipolar transistors are designed to
afford the greatest common-emitter current gain, ßdc or βF, in forward-active mode. If this
is the case, the collector–emitter current is approximately proportional to the base
current, but many times larger, for small base current variations.
 Saturation: With both junctions forward-biased, a BJT is in saturation mode and
facilitates high current conduction from the emitter to the collector. This mode
corresponds to a logical "on", or a closed switch.
 Cutoff: In cutoff, biasing conditions opposite of saturation (both junctions reverse
biased) are present. There is very little current, which corresponds to a logical "off", or an
open switch.
 Reverse-Active (or Inverse-Active or Inverted): By reversing the biasing conditions of
the forward-active region, a bipolar transistor goes into reverse-active mode. In this
mode, the emitter and collector regions switch roles. Because most BJTs are designed to

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maximize current gain in forward-active mode, the ßdc or βF in inverted mode is several
times smaller. This transistor mode is seldom used, usually being considered only for
failsafe conditions and some types of bipolar logic. The reverse bias breakdown voltage
to the base may be an order of magnitude lower in this region.
In Summary

Input or Output or Operation Device Function (The

S/No EB - Junction CB - Junction Region Transistor acts as)
1 Forward Forward Saturation Fully ON switch
2 Forward Reverse Active Amplifier, Oscillator
3 Reverse Reverse Cut - off Fully OFF switch
4 Reverse Forward Inverse Active Inverter

In most times for both PNP and NPN transistors, the emitter-base junction (EB) is forward-
biased while the collector-base junction (CB) is reverse-biased to use the device as an amplifier,
oscillator, mixer, detector, and so on. The working principle of NPN transistor is discussed here
and that of PNP transistor is similar except the fact that roles of free electrons and holes are
interchanged as well as current directions and biasing polarities are reversed.

Fig.3.2. Transistor Biasing Modes

In the NPN transistor (Fig.3.2 & Fig. 3.3), the EB junction is forward-biased by VBE, so that the
majority charge carriers (electrons) are emitted from the emitter into the base because the
negative potential of the battery of VBE repels the electrons from the N-type material (emitter).

The collector-base junction (CB) is reverse-based by VBC to collect or attract the most of emitted
electrons (say, about 99%) crossing the CB junction as collector current (IC). Some of the charge
carriers from the emitter, which do not reach the collector (say, about1%), entering the base
(recombination) and flow through the base back to the emitter.

This is a very small current and known as the base current (IB). Thus, the emitter current (IE) is
the total transistor current which is the sum of base current and collector current (IC).

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IE = IC + IB ------------------------------------------------ (3.1)

Since IB is very much small, IE and IC are almost equal(IC = IE). Where,  is fraction of emitter
current which flows to collector (0.98 - 0.998).
The small base current IB controls the much larger collector current IC. I C is proportional to IB. This
is generally known as the transistor effect.

Fig. 3.3: Transistor Operation & Direction of currents

In the case of a PNP transistor, holes will be drawn from the emitter into the base region by the
forward bias, and will then be pulled into the collector region by the higher negative bias

Since the CB junction is reverse-biased, a very small minority charge carrier, called
Reveres Saturation Current, flows through the junction. This current is termed as collector-base-
leakage-current (ICBO). ICBO means, current flowing form collector to the base when the emitter
junction is open. This is due to thermally generated electron-hole pairs even during normal
operation. We can now define another equation adding the effect of ICBO that indicates the total
collector current as:

IC = IE + ICBO. ------------------------------------------------------------------ (3.2)

Note! The circuit current flowing direction is opposite to the electrons (majority charge
carriers) drift direction, because of conventional current direction. (See Fig.3.2 &
Fig.3.3).

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3.4. Modes of BJT Configurations (Connections)

A transistor is a three terminal device. For applications such as amplifier circuit, four terminals
are required; two for the input section and two for the output section. So, one of the three
terminals of the transistor should be made common for both the input and for the output
terminals in such a case. Depending on which of the three terminals is used as common terminal,
there are three different configurations: common emitter (CE), common base (CB) and common
collector (CC). The common emitter (CE) is the most typical configuration:

1. Common - Base (CB) - The base is common for the input and the output (Fig.3.4b).

2. Common-Emitter (CE)-The emitter is common for the input & the output (Fig.3.4a)

3. Common - Collector (CC) - The collector is common for the input and the output (Fig.3.4c)

These modes are also known as, Grounded-base, Grounded-emitter and Grounded-collector.

Fig.3.4. Mode of BJT Configurations

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Transistor Current Gain

One of the important parameters of transistor configurations is the current amplification factor
(current gain).

Common - Base (CB);

The current amplification factor for CB mode is known as alpha () and is expressed as: -
I
Dc current gain (dc) = C ----------------------------- (3.3)
IE
Where IC and IE are the levels of current at the point of operation, αdc is fraction of emitter
current which flows to collector.

Ac current gain (ac)

Change in Output Current I C

 ac   , with VCB constant ----------- (3.4)
Change in Input urrent I E

The ac alpha (ac) is formally called the common-base (short-circuit) amplification factor.
Note that, αac and αdc are approximately equal and their values lie between 0.95 and 0.998.

Common - Emitter (CE)-

The current amplification factor for CE mode is called bat () and expressed as: -

IC
Dc current gain (ßdc) = ------------------------------------- (3.5)
IB
Where IC and IB are determined at a particular operating point on the characteristics
We have also another parameter from eq. 3.5

IC = dcIB ……………………………………… (3.6)

Ac current gain (ßac)

Change in Output Current I C

 ac   , with VCE = constant------ (3.7)
Change in Input Current I B

The formal name for ßac is common-emitter forward-current amplification factor.

Since the collector current is usually the output current for a common-emitter configuration
and the base current is the input current, the term amplification is included in the nomenclature
above.

Note again that βac = βdc = β or hFE = hfe

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Common - Collector (CC):

The current amplification factor for CC mode is also called beta prime () and given as: -

Dc current gain
I E I C  I B 
(dc) =     1 .................................................. (3.8a)
IB IB
Ac current gain (ac)

Change in Output Current I E

 ac'      1 , with VEC = constant ----------- (3.8b)
Change in Input Current I B

Note!  dc is always less than unity (one), and it is between 0.95 - 0.998.
 dc and dc are always much greater than unity.
 They are between dc = 19 - 500 & dc = 20 -500 and more.
 In data sheets ac is given as hfe

The relationship between  dc and  dc: -

From equation (3.1), IE = IC + IB.

Dividing throughout by IC we get,

IE I
 1  B -------------------------------- (3.9)
IC IC
As defined already the ratio of the collector current to the emitter current is dc = IC / IE. The
ratio of the collector current to the base current is ßdc = IC / IB. Making these substitutions in
equation (3.9) we get

1 1  
 1 . Simplifying, we get,   or   ------------- (3.10)
 dc  dc  1 1

Examples

1.       




   

Note! Examples 4 & 5 are typical and values.

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3.5. BJT Characteristic Curves

To study the behavior of a particular transistor, it is recommended that the relationship between
its (1) Input current and its input voltage and its (2) Output current and its output voltage should
be graphed and analyzed. This plotted graph is known as a characteristic curve of the device.
Thus, a transistor may have two characteristic curves known as input characteristic curves and
output characteristic curves respectively.

These characteristic curves are used to determine the important parameters of a transistor
graphically such as: -

 Current gain
 Input and output impedances
 Voltage gain

3.5.1. Common - Base Characteristic Curves

Input characteristic curves:

This is a plot of input voltage VEB versus input current IE for various values of output voltage
VCB as a constant parameter (Fig 3.5). As the forward bias VEB is increased, the input current IE
increases similar to diode characteristics. If VCB is increased, then IE increases slightly. This is
due to the increase in electric field aiding the flow of electrons from emitter.

Fig 3.5: CB Input and Output Characteristics

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Output Characteristic Curves:

It is plotted output voltage VCB versus output current IC for various values of input current IE as a
parameter (Fig 3.5 above). The three regions (active, cutoff and saturation) can be identified:

Active Region: Region to the right of y-axis, above IE = 0 mA curve, where the curves are linear.
IE is positive nonzero (i.e., E-B diode is forward biased) and VCB is positive (i.e., C-B diode is
reverse biased). When VCB is increased, IC increases slightly. This is because, when VCB is
increased, depletion region width at C-B junction increases, so effective base width decreases
and IB decreases. Hence IC increases. This effect is known as early effect (also called base width
modulation). If IE is increased, IC also increases and when IE = 0, IC = ICBO (reverse saturation
collector current in common Base with emitter Open). ICBO doubles for every 10 degree
centigrade rise in temperature.

Cutoff Region: Region below IE = 0 mA curve. Here IE is less than zero (E-B diode is reverse
biased) and VCB is positive (C-B diode is reverse biased) .The transistor is said to be in OFF state
since IC is zero or (IC = ICBO).

Saturation Region: Region to the left of y-axis, above IE = 0 mA curve. Here IE is positive
nonzero (E-B diode forward biased) and VCB is negative (C-B diode is forward biased) IC
decreases exponentially in this region. As shown in the curve, when VCB is reduced to zero, IC
still flows. This is because when VCB is zero, there is still a barrier potential which assists the
flow of IC. To stop the flow of the collector current (IC) or the flow of charge carriers, the
collector- to base (CB) junction has to be forward biased (less reverse biased).

Important Parameters:

a) Input Impedance (Resistance): Ratio of the change in VEB to corresponding change in IE, with
VCB held constant.
V EB
Z in  , with VCB constant………… (3.11)
I E
b) Output Impedance (Resistance): Ratio of the change in VCB to corresponding change in IC,
with IE held constant.
VCB
Zo  , with IE constant ----------------- (3.12)
I C
c) Current Gain: Ratio of the change in collector current to the change in emitter current, with
VCB held constant.
I C
 ac  , with VCB constant.
I E
d) Voltage Gain: Ratio of the change in output voltage VCB to the change in input voltage VEB
with IE constant
VCB
AV  , with IE constant ------------- (3.13)
VEB

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3.5.2. Common - Emitter Characteristic Curves

The emitter is common, the base is input terminal and the collector is output terminal. Thus, this
arrangement allows us to get two characteristics curves: namely; input characteristic curves and
output characteristic curves.

Input characteristic curves

This is a plot of input voltage VBE versus input current IB for various values of output voltage
VCE as a parameter (Fig 3.6). As the forward bias VBE is increased, the input current IB increases
similar to diode characteristics. If VCE is increased, then IB decreases slightly. This is due to early
effect.

Fig.3.6. CE Input and Output Characteristics

Output Characteristic Curves: -

It is a plot of output voltage VCE versus output current IC for various values of input current IB as
a parameter shown in Fig 3.6 above. Three regions (Active, Cutoff and Saturation) can be
identified again:

Active Region: Region to the right of VCESat, above IB = 0 curve, where the curves are linear.
Note that VCE = VCB + VBE (See Fig.3.4a). When VCE = 0, IC = 0.

If VCE > VCESat, then VCB becomes positive (i.e., C-B diode is reverse biased) VCESat is around
0.7V for silicon transistor. If IB > 0, then it means E-B diode is forward biased. When VCE is
increased, IC increases slightly due to early effect and remains almost constant, but not as much
constant as common base output characteristic. The slope of the common emitter is much more
pronounced than that of the common base output characteristic. This is because IE is not constant
as in a common base circuit.

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If IB is increased, IC also increases. Since the CE junction is reverse-biased when IB = 0, a very

small minority charge carrier, called Reveres Saturation Current, flows through the junction.
This current is termed as collector-emitter-leakage-current (ICEO). ICEO means, current flow
from collector to the emitter, when the base junction is open. This is due to thermally generated
electron-hole pairs even during normal operation. ICEO is much greater than ICBO of CB
configuration. Mathematically;

ICEO = (β + 1) ICBO----------------------------------------- (3.14)

Thus, when IB = 0, IC = ICEO. We can now define another equation including the effect of ICEO
that indicates the total collector current as:

IC = IE + ICEO = --------------------------------------------- (3.15)

Cutoff Region: Region below IB = 0 curve. Here E-B diode and C-B diode are both reverse
biased. Transistor is said to be in OFF state since IC is almost zero.

Saturation Region: Region to the left of VCESat and right of y-axis. Here E-B diode and C-B
diode are both forward biased and IC is at its maximum value.

Important Parameters:

(a). Input Impedance: Ratio of the change in VBE to corresponding change in IB, with VCE held
constant
VBE
Zi  , with VCE constant ----------------------- (3.16)
I B

(b). Output Impedance: Ratio of the change in VCE to corresponding change in IC, with IB held
constant.
VCE
ZO  , with IB constant----------------------------- (3.17)
I C

(c). Current Gain: Ratio of the change in collector current to the change in base current, with
VCE held constant.

I C
 ac  , with VCE constant--------------------------- (3.18)
I B

(d).Voltage Gain: Ratio of the change in output voltage to the change in input voltage with IB
held constant.

VCE
AV  , with IB constant -------------------------- (3.19)
VBE

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3.5.3. Common Collector Characteristics

Input characteristic curves:

It may be plotted VBC against IB with constant VEC. It is quite different from the other input
curves. IB reduces to zero as VBC increases at, VEC = VBC and thus, not practical.
IB (µA)

VCB = VEC VCB (V)

Fig.3.7. CC Input Characteristics
Output characteristic curves:

It is plotted VEC against IE with constant base current IB. Since IC is approximately equal to IE,
the common collector output characteristic curve is the same as that of the common emitter
circuit.
.
Important Parameters:

(a) Input Impedance: Ratio of the change in VBC to corresponding change in IB, with VEC held
constant.
VBC
Zi  , with VEC constant ----------------------- (3.20)
I B
(b) Output Impedance: Ratio of the change in VEC to corresponding change in IE, with IB held
constant.
VCE
ZO  , with IB constant------------------------- (3.21)
I E
(c) Current Gain: Ratio of the change in emitter current to the change in base current, with VEC
held constant.
I
 ac'  E   ac  1, with VCE constant--------------- (3.22)
I B
(d) Voltage Gain: Ratio of the change in output voltage to the change in input voltage with IB
held constant.
VCE
AV  , with IB constant --------------------------- (3.23)
V BE

Note again that: 1. β'ac = β'dc = β'

2. VBE, VCB and VCE are negative for PNP transistors.

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Reading Transistor Specific Data Sheets

When you look at a data sheet for a device (transistor), you should start with the maximum
ratings because there are the limits on the device current, voltage, power and other quantities
such as:
 Breakdown voltage ratings (VCE,VCB)
 Maximum current rating (IC max)
 Maximum power rating (PD max = IC max  VCE)
 Maximum and minimum current gain values (dc = hFE or ac = hfe)

Testing of Transistors

Transistors can be damaged by heat when soldering or by misuse in a circuit. If you suspect that
a transistor may be damaged it can be tested with an
ohmmeter:

Use an ohmmeter to check each pair of leads for

conduction. Set a digital multimeter to diode test and an
analogue multimeter to a low resistance range.

Test each pair of leads both ways (six tests in total):

 The base-emitter (BE) junction should behave
like a diode and conduct one way only.
 The base-collector (BC) junction should behave Fig.3.8. Testing an NPN transistor
like a diode and conduct one way only.
 The collector-emitter (CE) should not conduct either way.

The diagram shows how the junctions behave in an NPN transistor. The diodes are reversed in a
PNP transistor but the same test procedure can be used.

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3.6. Bipolar Junction Transistors Biasing Circuit Types

3.6.1. Introduction

One of the most common applications of transistors that should be stated repeatedly is its role in
amplifier circuits. For a faithful amplification we require that a transistor should be operated in
active region throughout the duration of input signal. To ensure this, proper dc voltages should
be applied which result to a situation called biasing.

The biasing of a transistor circuit is the selection of proper components and dc supplies. To
establish these dc operating conditions, the appropriate circuit operation must be obtained. Thus,
if this particular circuit dose not have the correct biasing, it will not operate properly.

For a Bipolar Junction Transistor to operate normally, it is essential that a dc voltage known as
base – to – emitter voltage (VBE), collector – to – base voltage (VBC) and collector –to - emitter
voltage (VCE) be maintained between its base, emitter and collector.

Applying these biasing voltages is of course, possible from dry cell batteries as shown in Fig.
3.2, 3.3 and 3.4. But this biasing method is most of the time very small to operate the transistor
properly and is also expensive. Thus, to obtain higher enough and appropriate biasing voltages,
BJTs are biased due to current flows through their base, emitter and collector resistors.
Therefore, bipolar junction transistors are said to be current controlled devices.

The main purpose of the dc biasing circuit is to set up the initial dc values of:
 Base current (IB)
 Collector current (IC)
 Collector – Emitter voltage (VCE)
from a single power source (supply) called VCC.

These initial dc values are called operating point of a device, also known as bias point, quiescent
point, or Q-point. This is a point on the output characteristics that shows the DC collector–
emitter voltage (Vce) and the collector current (Ic) with no input signal applied. This operating
point is expected to remain almost at the center of the active region of the device.

3.6.2. BJT DC Biasing Circuit Types

The most common five biasing circuits used in small signal bipolar transistor amplifiers are
discussed below:
1. Fixed Bias Circuit 2. Collector-to-Base Bias Circuit
3. Fixed Bias with Emitter resistor 4. Voltage Divider Bias
5. Emitter Bias

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3.6.2.1. Fixed Bias (Base Bias)

This form of biasing is also called base bias. In this biasing circuit, the base current (IB), remains
relatively constant (fixed) regardless of variations in the collector current (IC) by the collector
circuit bias (VCC) and the base resistor (RB). Since VCC, VBE and RB are constant; IB remains
constant at a particular level. Therefore, this type is called fixed bias type of circuit. Base bias
circuit is most useful in switching circuit.

Fig.3.9. Fixed Bias (Base bias)

Circuit Analysis
a) Input Section
In the given circuit (Fig.3.9), from KVL around the supply – base – ground circuit we get,
VCC = IBRB + VBE
Vcc  V BE
Solving for IB, IB = --------------------- (3.24)
RB
Vcc
Assuming VCC >>VBE, IB = for maximum IB values
RB
b) Output Section
Also for given circuit, applying the KVL around the supply –collector- ground circuit,
VCC = ICRC + VCE. Solving for VCE,

VCE = Vcc - ICRC ----------- (3.25)

The common-emitter current gain of a transistor is an important parameter in circuit design, and
is specified on the specific data sheet for a particular transistor denoted as β or hfe

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Because IC = βIB, we can obtain IC as well. In this manner, operating point given as (VCE, IC) can
be set for a given transistor. Thus, for VCE = 0V and VBE neglected,

Vcc Vcc
IB = ---------1 and IB = ---------2
RC RB

Solving for RB from equations 1 & 2, RB = βRC

Merits:
 It is simple to shift the operating point anywhere in the active region by merely changing
the base resistor (RB)
 Simple circuit; very small number of components are required.

Demerits:
 The collector current does not remain constant with variation in temperature or power
supply voltage. Therefore the operating point is unstable.
 When the transistor is replaced with another one, considerable change in the value of β
can be expected. Due to this change the operating point will shift.

Fig.3.10. Operating Point Shift Conditions

 For small-signal transistors (e.g., not power transistors) with relatively high values of β
(i.e., between 100 and 200), this configuration will be prone to thermal runaway.

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In particular, the stability factor, which is a measure of the change in collector current with
changes in reverse saturation current, is approximately β +1. To ensure absolute stability of
the amplifier, a stability factor of less than 25 is preferred, and so small-signal transistors
have large stability factors.

Usage (Application):
Due to the above inherent drawbacks, fixed bias is rarely used in linear circuits (i.e., those
circuits which use the transistor as a current source). Instead, it is often used in circuits where
transistor is used as a switch. However, one application of fixed bias is to achieve crude
automatic gain control in the transistor by feeding the base resistor from a DC signal derived
from the AC output of a later stage.
Examples:
A. Given VCC = 20V, β = 100, IC = 1mA, VCE = 10V and NPN Silicon transistor with VBE of
0.6V. Required: Design a CE fixed bias circuit.
Solution: 1. KVL for output section; VCC = ICRC + VCE

VCC  VCE 20V  10V

Solving for RC, RC =   10 K 
IC 1mA

2. From VCC = VRB + VBE, We get VCC = IBRB + VBE

VCC  V BE 20V  0.6V

Solving for RB, RB =   1 .9 M 
IB 10 A

Then RB is taken as 1.96MΩ (standard value)

B. For the Fixed Bias circuit (Fig.3.9), VCC = 20V, Rc = 2KΩ, RB = 270 KΩ, β = 75 and
Silicon transistor of 0.6V is given. Determine the transistor operating point (Q-point) values.

Solution:
1. For input section, from VCC = IBRB + VBE solving for IB,
IB = (VCC – VBE) / RB = (20V – 0.6V) / 270KΩ
= 19.4V / 270KΩ = 0.0718519mA  71.85 µA = IBQ

2. For output section, from Ic = βIB,

 Ic = 75 x 71.85 µA = 5.388mA  5.4mA = ICQ
 Solving for VCEQ, from Vcc = IcRc – VCE,
VCEQ = 20V – 5.4 x 10-3A x 2x10 3 Ω = 20V – 10.8V = 9.2V
So, the Q-point of the given transistor is (9.2V, 5.4mA)

Note! For a proper biasing condition, ICQ = ½ Icsat and VCEQ = ½ Vcc
Where, Icsat is the saturation (maximum) collector current determined as Icsat = VCC / Rc
Assignment: Is the above designed biasing circuit fulfilled this condition? If not how much
Q – Point shift is there from the center of active region of the transistor?

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3.6.2.2. Collector-to-Base Bias

This configuration employs negative feedback to prevent thermal runaway and stabilize the
operating point. In this form of biasing, the base resistor RB is connected to the collector instead
of connecting it to the DC source VCC. So any thermal runaway will induce a voltage drop across
RC resistor that will throttle (chock) the transistor's base current. Any change in VCE alters the
level of IB. The change in IB alters IC and tends this to return VCE towards its original value.

Fig.3.11. Collector-to-Base Bias

Circuit Analysis
a) Input Section
From KVL, Vcc = VRc + VRb +Vbe . Thus, the voltage across the base resistor Rb (VRb) is

From Ic = βIb, and so

From Ohm's law, the base current , and so

Hence, the base current Ib is

------------------------------------------- (3.26)

VCC  V BE
Note! For IB << βIB, Equation (3.26) be comes IB =
R B   RC

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b) Output Section
IC = βIB
From Vcc = IcRc +VCE,
VCE = Vcc – IcRc = Vcc - βIBRC ------------------------------------------ (3.27)
If VBE is held constant and temperature increases, then the collector current IC increases.
However, a larger IC causes the voltage drop across resistor RC to increase, which in turn reduces
the voltage across the base resistor RB, (VRB). A lower base-resistor voltage drop reduces the
base current IB, which results in less collector current IC. Because an increase in collector current
with temperature is opposed, the operating point is kept stable.

Merits:
 Circuit stabilizes the operating point against variations in temperature and β (ie.
replacement of transistor)

Demerits:
 In this circuit, to keep Ic independent of β, the following condition must be met:

Which is the case when

 As β-value is fixed (and generally unknown) for a given transistor, this relation can be
satisfied either by keeping RC fairly large or making RB very low.
 If RC is large, a high VCC is necessary, which increases cost as well as precautions
necessary while handling.
 If RB is low, the reverse bias of the collector–base region is small, which limits
the range of collector voltage swing that leaves the transistor in active mode.
 The resistor RB causes an AC feedback, reducing the voltage gain of the amplifier. This
undesirable effect is a trade-off for greater Q-point stability.

Usage (Application):
The feedback also decreases the input impedance of the amplifier as seen from the base, which
can be advantageous. Due to the gain reduction from feedback, this biasing form is used only
when the trade-off for stability is warranted.

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3.6.2.3. Fixed Bias with Emitter Resistor

Fixed Bias with Emitter Resistor, also called Emitter Feedback Bias, is a fixed bias circuit
modified by attaching an external resistor to the emitter. This resistor introduces negative
feedback that stabilizes the Q-point. This is due to that the emitter current (Ie) flowing through
the emitter resistor (Re) is equal to Ic + Ib. The collector current is brought from the output
circuit back to the input. This current helps to stable the circuit operation.

Fig.3.12. Fixed Bias with Emitter Resistor

Circuit Analysis
a) Input Section
Applying KVL to the circuit loop,
Vcc = IBRB + VBE + IERE
Thus, the voltage across the base resistor is
VRB = VCC – IERE - VBE.
From Ohm's law, the base current is

V RB
IB = ----------------------------- (3.28)
RB

The way feedback controls the bias point is as follows. If VBE is held constant and temperature
increases, emitter current increases. However, a larger IE increases the emitter voltage VE = IERE,
which in turn reduces the voltage VRB across the base resistor. A lower base-resistor voltage drop
reduces the base current, which results in less collector current because IC = ßIB. Collector
current and emitter current are related by IC = α IE with α ≈ 1, so increase in emitter current with
temperature is opposed, and operating point is kept stable.

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Similarly, if the transistor is replaced by another, there may be a change in IC (corresponding to

change in β-value, for example). By similar process as above, the change is negated and
operating point kept stable.
For the given circuit, from the input section KVL equation,

Vcc = IBRB + VBE + IERE = VBE + IBRB + (IC +IB) RE

= VBE +IBRB + IB (β+1) RE
= VBE +IB [RB + (β+1) RE]

VCC  V BE
Solving for IB, IB = ------------------------------------ (3.29)
RB    1RE 
b) Output Section
1. IC = βIB
2. From Vcc = VCE + IcRc + IERE,
VCE = Vcc – (IcRc + IERE) For Ic  IE,

VCE = Vcc - (IcRc + IcRE)

And thus, VCE = Vcc – Ic (Rc + RE) -------------- (3.30)
Merits: The circuit has the tendency to stabilize operating point against changes in temperature
and β-value.
Demerits: In this circuit, to keep IC independent of β the following condition must be met:

Which is approximately the case if, (β + 1 ) RE >> RB.

 As β-value is fixed for a given transistor, this relation can be satisfied either by keeping
RE very large or making RB very low.
 If RE is of large value, high VCC is necessary. This increases cost as well as
precautions (safety measures) necessary while handling. If RB is low, a separate
low voltage supply should be used in the base circuit. Using two supplies of
different voltages is impractical.
 In addition to the above, RE causes ac feedback which reduces the voltage gain of the
amplifier.

Usage: (Application):
The feedback also increases the input impedance of the amplifier when seen from the base,
which can be advantageous. Due to the above disadvantages, this type of biasing circuit is used
only with careful consideration of the trade-offs involved.

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3.6.2.4. Collector-Stabilized Biasing

a) Voltage Divider Bias

The voltage divider is formed using external resistors R1 and R2. The voltage across R2 forward
biases the emitter junction. By proper selection of resistors R1 and R2, the operating point of the
transistor can be made independent of β. In this circuit, the voltage divider holds the base voltage
fixed independent of base current provided the divider current is large compared to the base
current. However, even with a fixed base voltage, collector current varies with temperature (for
example) so an emitter resistor is added to stabilize the Q-point, similar to the above circuits with
emitter resistor.

Fig.3.13.Voltage Divider Bias

Circuit Analysis
Approximate Analysis
Input Section
In this circuit the base voltage VB = VR2 is determined first as: VB = (VCC x R2) / (R1 + R2) or

voltage across ------------------- (3.31)

provided .
Also ------------------- (3.32)
For the given circuit IB may be calculated as:

--------------------------- (3.33)

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Output Section
From the equation (3.32), IE is determined and is taken as ICQ

V B  V BE
IE =  I CQ -------------------------------------- (3.34)
RE

Solving for VCEQ, VCEQ = Vcc – IcRc - IERE = Vcc – IE (RC + RE)

VCEQ = Vcc – IcQ (RC + RE) --------------------- (3.35)

IE
From IE = IB (β +1), IB can also be calculated as: IB =
 1
Merits:
 Like above circuits, only one dc supply is necessary.
 Operating point is almost independent of β variation (see Eqs.3.32, 3.34 and 3.35).
 Operating point stabilized against shift in temperature.
Demerits: In this circuit, to keep IC independent of β the following condition must be met:

This is approximately the case if

Where R1 || R2 denotes the equivalent resistance of R1 and R2 connected in parallel.

 As β-value is fixed for a given transistor, this relation can be satisfied either by keeping
RE fairly large or making R1||R2 very low.
 If RE is of large value, high VCC is necessary. This increases cost as well as
precautions necessary while handling.
 If R1 || R2 is low, either R1 is low, or R2 is low, or both are low. A low R1 raises
VB closer to VC, reducing the available swing in collector voltage, and limiting
how large RC can be made without driving the transistor out of active mode. A
low R2 lowers Vbe, reducing the allowed collector current. Lowering both resistor
values draws more current from the power supply and lowers the input resistance
of the amplifier as seen from the base.
 AC as well as DC feedback is caused by RE, which reduces the AC voltage gain of the
amplifier. A method to avoid AC feedback while retaining DC feedback is discussed
below.

Usage (Application):
The circuit's stability and merits as above make it widely used for linear circuits.

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Exact Analysis
A more exact analysis of the Voltage Divider Biasing circuit than the above method can be
obtained by applying Thevenin's theorem to the circuit.

Determination of Thevenin voltage (VTh): To find the Thevenin voltage, remove the transistor
base circuit to the voltage divider network of the points A and B in Fig.3.13. Then by the voltage
divider rule the potential across R2, which is the Thevenin voltage is:
V R
V R 2  VTh  CC 2 ........................................ (3.36)
R1  R2
Determination of the Thevenin resistance (RTh): To find the Thevenin resistor, short circuit the
source voltage (VCC). Then the two resistors, R1 and R2 will be in parallel as shown in Fig.3.14.
and the Thevenin resistance is the equivalent of the two resistances R1 and R2 in parallel. Thus

R1 R2
RTh  ................................... (3.37)
R1  R2

Source short R1
A A A

R1 R2
R2  R1 R2  RTh 
R1  R2
B B B

Fig.3.14. Determination of the Thevenin resistance (RTh)

The Thevenin equivalent of the circuit in Fig.3.13 will be

+ VCC

IC RC

RTh
VCE
IB
+ VBE
VTh
- IE RE

Fig.3.15.Thevenin equivalent of Fig.3.13

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Analysis of the circuit in Fig.3.15.

Input Section

Applying the KVL to the input circuit we get

VTh  RTh I B  VBE  RE I E ............................. (3.38). Now substituting IE = (ß +1) IB,

VTh  RTh I B  VBE  RE   1I B . Solving for IB, it yields;
VTh  V BE
IB  .......................... (3.39a). Since VBE is small compared to VTh and
RTh    1R E
beta is large compared to 1, we can write;
VTh
IB  ................................... (3.39b).
RTh   R E
Output Section

The collector current is given by;

VTh
I C  I B  ....................... (3.40). Applying KVL to the output circuit we get;
RTh  R E
VCC  I C RC  VCE  I E RE .................... (3.41a). Taking IC is nearly equal to IE and solving
for VCE;
VCE  VCC  I C RC  RE  ...................... (3.41b).

In this case, equations (3.40) and (3.41b) together determine the operating point.

b) Voltage Divider with AC emitter bypass capacitor

The standard voltage divider circuit discussed above faces a drawback - AC feedback caused by
resistor RE reduces the gain. This can be avoided by placing a bypass capacitor (CE) in parallel
with RE, as shown in circuit diagram (Fig.3.16).

Fig.3.16.Voltage Divider with Emitter Bypass Capacitor

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This capacitor is usually chosen to have a low enough reactance at the signal frequencies of
interest such that RE is essentially shorted at AC, thus grounding the emitter. Feedback is
therefore only present at DC to stabilize the operating point, in which case any AC advantages of
feedback are lost. Of course, this idea can be used to shunt only a portion of RE, thereby retaining
some AC feedback.

3.6.2.5. Emitter Bias

When a split supply (dual power supply) is available, this biasing circuit is the most effective,
and provides zero bias voltage at the emitter or collector for load. The positive supply VCC is
used to reverse-bias the collector junction.

Fig.3.17. Emitter Bias

The negative supply VEE is used to forward-bias the emitter junction through RE. Only two
resistors are necessary for the common collector stage and four resistors for the common emitter
or common base stage.

Circuit Analysis
We know that, VB - VE = VBE.
If RB is small enough, base voltage (VB) will be approximately zero.

V EE  V BE
Therefore, emitter current (IE) is determined as IE =  I CQ
RE
RB
The operating point is independent of β if RE >>

Merit:
Good stability of operating point similar to voltage divider bias.

Demerit:
This type can only be used when a split (dual) power supply is available. Thus, seldom practical

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3.6.3. The DC Load Line

The DC load line is a graph that can be drawn on the output characteristic curve of a transistor to
represent all the possible pairs of the output current through the transistor (IO = IC) and the output
voltage across the transistor (VO = VCE) for a given amplifier dc circuit (See Fig.3.10). This IC
and VCE corresponding point is called an operating point or a quiescent point or just Q-point of
the transistor. The values of IC and VCE at this point are known as operating point values (ICQ and
VCEQ). Quiescent means the dc biasing condition of the given transistor circuit when there is no
ac input signal is applied.

VCC
IC / mA IC = IC (sat) =
RC

B
Q- Point 1
ICQ 1

Q-Point 2
ICQ 2 Q-Point 3

ICQ 3 DC Load Line

VCE = VCE (Off) = VCC

VCE / V
VCEQ 1 VCEQ 2 VCEQ 3 A

Fig.3.18.The DC Load Line

Drawing the DC Load Line

Taking the Fixed Bias (Base Bias) circuit (Fig.3.9.) as an example, for the KVL of the output
section, VCC = ICRC + VCE. This equation is called the DC load line equation. The DC load line
has two end points.
1. When IC = 0, (x- axis), that indicates VCC = VCE = VCE(off) Point A
2. When VCE = 0, (y-axis), that indicates IC = VCC / RC = IC(sat) Point B
By joining these two points, we get the Dc load line. The end points of the Dc load line are
leveled as:

IC(sat) or saturation current and VCE(off) or cut-off voltage

Note! A proper biasing condition for small signal amplifiers means, setting the operating point of
the given amplifier circuit at the center (middle) of the DC load line (active region) on the
output characteristic curve. In other words; ICQ = 0.5IC (sat) and VCEQ = 0.5VCC.

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3.7. Small Signal BJT Amplifiers and Parametric Representations

3.7.1. Introduction
An amplifier is a circuit using an active device such a transistor or an integrated circuit (IC) to
increase the intensity (strength) of current, voltage or power of a signal without changing the
shape of the waveform and the frequency.

Thus, amplification means the process of increasing the intensity (amplitude) of a signal.

Amplifiers are necessary in most applications because the desired signal is usually too weak to
directly useful. As an example, audio output from a microphone may be as little as one millivolt,
whereas the loudspeaker needs at least a few volts of audio signal. With an amplifier however, a
faint whisper can be made to fill a large room with a very loud sound.

Two port networks (system) are widely used to model transistors amplifier circuit blocks.

Fig.3.19. A Typical Two Port Amplifier System

Depending on the input and output signal levels, amplifiers may be classified as:

Small signal and Large signal Amplifiers. Small signal BJT amplifiers are discussed in this
chapter and Large Signal Amplifiers will be covered in chapter 7.

As far as we are concerned with transistors application as small signal amplifiers, the most
common transistor amplifier parameters involved are defined below.

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3.7.2. Parameters of an Amplifier

Parameters are quantities (factors), those decide the performance of an amplifier.
The three most important parameters of any amplifier are: -

1. Input Impedance(Zi): or ac resistance

Change in Input Voltage Vi
Zi   Vo held cons tan t
Change in Input Current I i
2. Output Impedance (Zo): or ac output resistance ro

output voltage Vo Vo

Z o or ro    I in  cons tan t
output current I o I o

3. Amplification factor (Gain): determines how much the input signal is increased.
OUTPUT SIGNAL
Generally: - Gain (A) =
INPUT SIGNAL
a condition
The gain can be further subdivided as
 Voltage gain (Av)
 Current gain (AI)
 Power gain (Ap)
out put voltage Vo V
3.1. Voltage Gain: (AV) =   o
input voltage VI Vin
Io held constant
Thus, the voltage gain is the ratio of the output voltage to the input
voltage with constant output current

output current I o
3.2. Current Gain: - (AI)  Vo  cons tan t
input current II
The ratio of output current to the input current with constant output
voltage
output power
3.3. Power Gain: - (AP) =  AV x AI
input power
The ratio of output power to the input power or the product of a voltage
gain and a current gain of each stage is a power gain.

Further more refer to section 3.5 of this chapter how these parameters can be determined.

Note! A negative sign of a linear gain indicates a 1800 phase shift between the input and the
output signals.
In most circuits, active devices such as, Transistors and ICs are used as a main amplifier part.
In addition, resistors, inductors and capacitors are required to form a complete amplifier circuit.
These passive components provide paths for the input and output signals.

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Example of a Typical Small Signal Amplifier Circuit

Fig.3.20. A Typical Small Signal Amplifier Circuit

Purpose of Each Component

1. (VS + RS): - a signal source as the main input including its internal resistance.

2. C1: - the input coupling capacitor, used to couple or pass the incoming input signal and
block a dc voltage from being applied to succeeding (next) stage.

3. R1 & R2: - voltage divider network, to develop bias voltage to the base.

4. Transistor: - an active device uses as a main amplifier component.

5. RC: - a collector resistor, which determines the voltage to be applied to the collector by
dropping some of the VCC (main supply dc voltage).

6. RE: - an emitter resistor, which develops emitter bias voltage and also used
as a temperature stabilizer (See Fig.3.16.).

7. C3: - a bypass capacitor (CE), which grounds any unwanted ac ripples.

8. C2: - an output coupling capacitor that used to pass (transfer) the amplified output signal to
the load impedance or to the next stage. It also blocks a dc voltage not to be coupled.

9. RL: - a load resistor, which develops the output ac voltage across it (if it is connected).

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The above described amplifier circuit is known as a Common – Emitter

In a common-emitter amplifier the input signal is applied to the base and the output is taken from
the collector.
A common-emitter amplifier is the most commonly used type for two reasons

1. It has high voltage and power (current) gain.

2. Its input and output impedances (ac resistances) are stable or moderate.

Other common configuration (Circuits)

a) Common – Base b) Common - Collector or (Emitter Follower)

i) The input signal is applied to the emitter i) The input signal is applied to the base
and the output is taken from the collector. and the output is taken from the emitter.
ii) Without C3 (CE) ii) Without C3 (CE)
iii) RC is very low (can be neglected)
iv) C2 couples RE and RL

Comparisons of CB, CE, and CC Amplifier Circuits:

Parameter or
S/N CB CE CC
Characteristics
Input impedance Low  20-40 Moderate High = 0.1-5M
1
(Zin)  200-1500
Output impedance High  0.9-1.5M Moderate Low = 100-500
2
(ZO) 30-100K
3 Voltage gain (AV) High High Low 1
4 Current gain (AI) Low  () High () = 19 High () 20
5
Power gain (Ap) High  AVb V. High High 
(Av x AI)
Phase shift b/n No = 00 Yes = 1800 No = 00
6
Vin & Vo
1 Isolation input & output Universal voltage Impedance matching,
signals isolate Amp. due to a high Zo circuit with
2. Impedance matching, a  Has high AV, AI a low Zin circuit.
low Zo circuit with a & AP Ex CB circuit
7 high Zin circuit.  Its Zin & Zo are
Application Ex .CC circuit moderate (stable)
 It is good for
cascading two
CE circuits.

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3.7.3. Analysis of Small Signal BJT Amplifier Circuits

The simplest way to analyze the circuit is to split it into two parts as;

 DC Analysis
 AC Analysis

The DC Analysis; is to obtain or drive a dc equivalent circuit of the given amplifier circuit as an
example of Fig.3.20. This dc equivalent circuit is used to determine the dc operating values. The
dc equivalent of a network is obtained by:

1. Setting all ac sources to zero and replacing them by an open-circuit equivalent

2. Replacing all capacitors by an open-circuit equivalent
3. Removing all elements bypassed by the open-circuit equivalents introduced by steps 1 and 2
4. Redrawing the network in a more convenient and logical form
5. Calculate all the dc operating values

The dc biasing of the device was then examined in detail in section 3.6. The dc equivalent circuit
for Fig.3.20 was driven as Fig.3.13.

We now begin to examine the small-signal ac response of the BJT amplifier by reviewing the
models (ac equivalent circuits) most frequently used to represent the transistor in the sinusoidal
ac domain.

The AC Analysis: is to obtain or drive the ac equivalent circuit of the given amplifier circuit as
an example of Fig.3.20. This ac equivalent circuit is used to determine the possible parameters
(ac values) of the amplifier. In summary, the ac equivalent of a network is obtained by:

1. Setting all dc sources to zero and replacing them by a short-circuit equivalent

2. Replacing all capacitors by a short-circuit equivalent
3. Removing all elements bypassed by the short-circuit equivalents introduced by steps 1 and 2
4. Redrawing the network in a more convenient and logical form
5. Calculate all necessary ac values (parameters)

Fig.3.21.Equivalent Circuit of Fig.3.20.redrawn for small-signal ac analysis.

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Let us further examine Fig. 3.20.and identify the important quantities to be determined for the
system. Since we know that the transistor is an amplifying device, we would expect some
indication of how the output voltage Vo is related to the input voltage Vi which defines the
V 
voltage gain  o  . Note also in Fig. 3.20, for this configuration that Ii = Ib and Io = Ic, which
 Vi 
I 
defines the current gain  o . The input impedance Zi and output impedance Zo will prove
 Ii 
particularly important in the analysis to follow.

There are two models (equivalent circuits) commonly used in the small-signal ac analysis of
transistor networks: the re equivalent model and the hybrid equivalent model.

re equivalent circuit ac analysis

Fig.3.22. re equivalent model for the common-emitter transistor configuration (say Fig.3.20)

a) Input Impedance (Zi)

In this configuration, the base current is the input current while the output current is Ic.
Since IC = IB, the current through the BE-junction diode (Ie) is therefore determined by

Ie = Ic + Ib = Ib + Ib = ( + 1)Ib

However, since the ac beta is typically much greater than 1, Ie  Ib.

(a) (b)
Fig.3.23. Determining Zi using the approximate model (a) and Impact of re on input impedance
the approximate model (b).

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The input impedance is determined by the following ratio:

Vi Vbe
Zi  
Ii Ib

The voltage Vbe is across the BE- junction diode resistance (re). The level of re
is still determined by the dc current IE. Using Ohm’s law gives (re = rd =26mV/ IE)

Vbe I b re
Vi = Vbe = Ie re  Ibre , Thus, Zi = 
Ib Ib
Zi  re -------------------------------------------- (3.42)

But for the CE voltage divider ac equivalent circuit, the input impedance is in parallel with the
voltage divider resistors. Thus,

Zi  rb║re ------------------------- (3.43) for low value of rb and Where rb = R1║R2

For the majority of situations rb is greater than re by more than a factor of 10, which is
permitting the approximation of equation 3.42

For the common-emitter configuration, typical values of Zi defined by re range from a few hundred ohms to the
kilohm range, with maximums of about 6–20 KΩ.

b) Output Impedance (Zo)

Recall that the output impedance of any system is defined as the impedance Zo determined when
Vi = 0. For Fig.3.21, when Vi = 0, Ii = Ib = 0, resulting in an open-circuit equivalence for the
current source. The result is:

Zo = RC║ro ---------------------------------------------- (3.44)

If ro > 10 RC, the approximation RC║ro  RC is frequently applied and

Zo = RC -------------------------- (3.45), for ro >10RC and for RL is open

Or
Zo = rc -------------------------- (3.46), for rc = RC║RL

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Applied Electronics I Bipolar Junction Transistors 36

c) Voltage Gain (Av)

The resistors ro and RC are in parallel (see Fig.3.21 and Fig.3.22).

Vo = - Io (RC║ ro) = - (Ib) (RC║ro) and Vi = IiZi = Ibre

Thus Av = Av 
Vo

I b RC ro    RC ro ----------------- (3.47a), and
Vi I b re re
Vo R
Av    C ----------------------------- (3.47b), for ro > 10RC and RL is open
Vi re
Or
VO rc
AV   ----------------------------- (3.47c) for rc = RC║RL
Vi re

Note! A negative sign of a linear gain indicates a 1800 phase shift between the input and the
output voltage signals

d) Current Gain (Ai)

I o I c I b
Ai    
Ii Ib Ib
So, Ai   ------------------------------------ (3.48a)

The current gain may also be determined as

Vo
Io RC  Vo  Z i  re
Ai        Av
Ii Vi  Vi  RC  RC
Zi
re
Thus, Ai  Av ----------------------- (3.48b)
Rc

e) Power Gain (Ap)

Po Vo I o
Ap    Av  Ai ------------ (3.49)
Pi Vi I i

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Applied Electronics I Bipolar Junction Transistors 37

hybrid equivalent circuit ac Analysis

Fig3.24.Two-Port hybrid model

The hybrid model is taken as the most suitable for modeling transistors. This model could be stated
by using two linear equations.

Vi = h11Ii + h12Vo -------------------------------------------------------------- (3.50a)

Io = h21Ii + h22Vo ------------------------------------------ (3.50b)

The parameters relating the four variables are called h-parameters from the word “hybrid.” The
term hybrid was chosen because the mixture of variables (V and I) in each equation results in a
“hybrid” set of units of measurement for the h-parameters. A more clear understanding of what
the various h-parameters represent and how we can determine their magnitude can be
developed by isolating each and examining the resulting relationship.

If we arbitrarily set Vo = 0 (short circuit the output terminals) and solve for h11 in Eq. (3.50a), the
following will result:
V
h11  i Vo  0 ------------------------------------------- (3.51)
Ii
The subscript 11 of h11 defines the fact that the parameter is determined by a ratio of quantities
measured at the input terminals and it is called the short-circuit input-impedance parameter.
Here we see that Ii and Vo are independent and Vi and Io are dependent variables.

Now, if Ii is set equal to zero by opening the input leads, we can solve for h12 as:
V
h12  i Ii  o --------------------------------------- (3.52)
Vo
It has no unit, since it is a ratio of voltage levels and is called the open-circuit reverse transfer
voltage ratio parameter. The subscript 12 of h12 reveals that the parameter is a transfer quantity
determined by a ratio measured quantity to appear in the numerator; the second integer defines
the source of the quantity to appear in the denominator. The term reverse is included because the
ratio is an input voltage over an output voltage rather than the reverse ratio typically of interest
of input to output measurements. The first integer of the subscript defines the measured quantity
to appear in the numerator and the second integer defines the source of the quantity to appear in
the denominator.

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Applied Electronics I Bipolar Junction Transistors 38

If in Eq. (3.50b) Vo is equal to zero again by shorting the output terminals, it will result for h21:
I
h21  o Vo  0 ------------------------------- (3.53)
Ii
Note that we now have the ratio of an output quantity to an input quantity. The term forward will
now be used rather than reverse as indicated for h12. The parameter h21 is the ratio of the output
current to the input current with the output terminals shorted. This parameter, like h12, has no
unit since it is the ratio of current levels. It is formally called a short - circuit forward transfer
current ratio parameter. The subscript 21 again indicates that it is a transfer parameter with the
output quantity in the numerator and the input quantity in the denominator.

The last parameter, h22, can be found by again opening the input leads to set Ii = 0 and solving
for h22 in Eq. (3.50b):
I
h22  o Ii o ------------------------------ (3.54)
Vo
Since it is the ratio of the output current to the output voltage, it is the output conductance
parameter and is measured in siemens (S). It is called the open-circuit output admittance
parameter. The subscript 22 reveals that it is determined by a ratio of output quantities.

Common names given to these new parameters when we apply the hybrid equivalent model to
transistors are given below.

h11 = hi h12 = hr h21 = hf and h22 = ho = 1 / ro

h11 input resistance hi h21 forward transfer current ratio hf

h12 reverse transfer voltage ratio hr h22 output conductance ho

Finally we can model our transistor as indicated in the following figure. This model is common
to any types of configurations discussed earlier.

Fig.3.25. Complete hybrid equivalent circuit

Fig.3.22. is redrawn below with the block transistor replaced by the detailed model discussed
above. We use the exact method if ho or Ro is mentioned otherwise we use the approximate
method. So, the analysis generally depends on the following general model.

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Applied Electronics I Bipolar Junction Transistors 39

hi
+ +
Rs
-
Vi ho RL
hrVo AC hf Ii AC Vo
AC
+
- -

Fig.3.26.The hybrid equivalent model as small signal amplifier

Current Gain:

Io
Ai  , But, Io = hf Ii + Voho. See Eq. (3.50a) ------------- (1) and also
Ii
Vo
Io = and we get , Vo   I o R L ------------. (2)
RL
Thus, Io = hf Ii + (-IoRL) ho or I o  I o R L ho  h f I i , which gives, I o 1  R L ho   h f I i
Finally we see that,

Io hf
Ai   …………… (3.55a) exact value
I i 1  RL ho
If RLho is very small due to the value of ho, Eq. (3.55a) is reduced to

Ai = hf…………………………………………… (3.55b), approximation

Voltage Gain:

Vo
Av  , But Vi  hi I i  hrVo
Vi
Io V
Substituting I i  1  ho R L  and I o   o from the above relations,
hi RL
 1  ho RL hi V
Vi  hi I i  hrVo  Vo  hrVo . Solving for the ratio O yields
h f RL Vi
Vo  h f RL
Av   .................................3.56a  exact
Vi hi  hi ho  hr h f RL

For hi ho  hr h f RL  hi ,

 h f RL
AV  ............................................................... 3.56b  appriximation
hi

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Applied Electronics I Bipolar Junction Transistors 40

Input Impedance:

Vi
Zi  , But for the input circuit, Vi  hi I i  hrVo . Substitute Vo   I o RL , we have
Ii
Io
Vi  hi I i  hr RL I o . Since Ai  , then I o  Ai I i .Thus, the equation for Vi be comes,
Ii
V
Vi  hi I i  hr RL Ai I i  I i hi  hr RL Ai  . Solving for the ratio i , we obtain
Ii
V hf
Z i  i  hi  hr R L Ai . Substituting Ai  ,
Ii 1  ho RL
V h f hr RL
Z i  i  hi  ................................. (3.57a) as exact value
Ii 1  ho RL
h f hr RL
In this case, for  hi ,
1  ho RL
Z i  hi ................................................ (3.57b) most familiar form of Zi (approximation)

Output Impedance:

Vo
Zo  , with the signal Vs set to zero. Thus for the input circuit Vs = 0,
I0
hV
I i   r o . Substituting this relationship into the output circuit equation we get,
R S  hi
h f hrVo V
I o  h f I i  hoVo    hoVo . Solving for the ratio o ,
RS  hi I0
V 1
Zo  o  . .......................................... (3.58a) exact value
Io h f hr
ho 
RS  hi
h f hr
In this case, for ho  , the output impedance of a transistor will reduce to the most
RS  hi
approximation form of
1
Zo  .............................. (3.58b) approximation
ho
Power Gain:

This is simply the product of voltage and current gains

h f RL hf h 2f RL
AP  AV  AI     ……………… (3.59) approximation
hi 1  hO RL hi 1  ho RL 

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